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doi: 10.3934/dcdsb.2021291
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Boundedness of the complex Chen system

1. 

Department of Mathematics, Shandong University, Weihai, Shandong 264209, China

2. 

Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China

Received  May 2021 Revised  October 2021 Early access December 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11701328) and Young Scholars Program of Shandong University, Weihai (No. 2017WHWLJH09)

Some ultimate bounds are derived for the complex Chen system.

Citation: Xu Zhang, Guanrong Chen. Boundedness of the complex Chen system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021291
References:
[1] C. Bailly and G. Comte-Bellot, Turbulence, Springer International Publishing Switzerland, 2015.  doi: 10.1007/978-3-319-16160-0.  Google Scholar
[2]

R. Barboza, On Lorenz and Chen systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850018, 8 pp. doi: 10.1142/S0218127418500189.  Google Scholar

[3]

R. Barboza and G. Chen, On the global boundedness of the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.  Google Scholar

[4]

S. Celikovsky and G. Chen, Generalized Lorenz systems family revisited, Int. J. Bifurcation Chaos, 31 (2021), 2150079, 15 pp. Google Scholar

[5]

D. Cheban and J. Duan, Recurrent motions and global attractors of nonautonomous Lorenz systems, Dyn. Syst., 19 (2004), 41-59.  doi: 10.1080/14689360310001624132.  Google Scholar

[6]

G. Chen, Generalized Lorenz systems family, preprint, arXiv: 2006.04066, 2020. Google Scholar

[7]

G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.  Google Scholar

[8] P. A. DavidsonY. Kaneda and K. R. Sreenivasan, Ten Chapters in Turbulence, Cambridge Univ. Press, 2013.   Google Scholar
[9]

A. C. FowlerJ. D. Gibbon and M. J. McGuinness, The real and complex Lorenz equations and their relevance to physical systems, Phys. D, 7 (1983), 126-134.  doi: 10.1016/0167-2789(83)90123-9.  Google Scholar

[10]

A. C. FowlerM. J. McGuinness and J. D. Gibbon, The complex Lorenz equations, Phys. D, 4 (1981/82), 139-163.  doi: 10.1016/0167-2789(82)90057-4.  Google Scholar

[11]

M. Franz and M. Zhang, Suppression and creation of chaos in a periodically forced Lorenz system, Phys. Rev. E, 52 (1995), 3558-3565.  doi: 10.1103/PhysRevE.52.3558.  Google Scholar

[12]

H. Haken, Analogy between higher instabilities in fluids and lasers, Phys. Lett. A, 53 (1975), 77-78.  doi: 10.1016/0375-9601(75)90353-9.  Google Scholar

[13]

C. Lainscsek, A class of Lorenz-like systems, Chaos, 22 (2012), 013126, 5 pp. doi: 10.1063/1.3689438.  Google Scholar

[14] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959.   Google Scholar
[15]

G. A. LeonovA. I. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656.  doi: 10.1002/zamm.19870671215.  Google Scholar

[16]

G. A. Leonov and N. V. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen, and Lu systems, Appl. Math. Comput., 256 (2015), 334-343.  doi: 10.1016/j.amc.2014.12.132.  Google Scholar

[17]

C. Letellier, G. F. V. Amaral and L. A. Aguirre, Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models, Chaos, 17 (2007), 023104, 11 pp. doi: 10.1063/1.2645725.  Google Scholar

[18]

D. LiJ. LuX. Wu and G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J. Math. Anal. Appl., 323 (2006), 844-853.  doi: 10.1016/j.jmaa.2005.11.008.  Google Scholar

[19]

X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization (in Chinese), Sci. China Ser. E, 34 (2004), 1404-1419.   Google Scholar

[20]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[21]

J. LüG. ChenD. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X.  Google Scholar

[22]

G. M. MahmoudT. Bountis and E. E. Mahmoud, Active control and global synchronization of the complex Chen and Lü systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4295-4308.  doi: 10.1142/S0218127407019962.  Google Scholar

[23]

J. Pedlosky, Finite-amplitude baroclinic waves with small dissipation, J. Atmos. Sci., 28 (1971), 587-597.  doi: 10.1175/1520-0469(1971)028<0587:FABWWS>2.0.CO;2.  Google Scholar

[24]

J. Pedlosky, The effect of $\beta$ on the chaotic behavior of unstable baroclinic wave, J. Atmos. Sci., 38 (1981), 717-731.   Google Scholar

[25]

W.-X. Qin and G. Chen, On the boundedness of solutions of the Chen system, J. Math. Anal. Appl., 329 (2007), 445-451.  doi: 10.1016/j.jmaa.2006.06.091.  Google Scholar

[26]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

[27]

H. Saberi NikS. Effati and J. Saberi-Nadjafi, New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system, J. Complexity, 31 (2015), 715-730.  doi: 10.1016/j.jco.2015.03.001.  Google Scholar

[28]

R. SaravananO. NarayanK. Banerjee and J. K. Bhattacharjee, Chaos in a periodically forced Lorenz system, Phys. Rev. A, 31 (1985), 520-522.  doi: 10.1103/PhysRevA.31.520.  Google Scholar

[29]

E. A. Sataev, Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Mat. Sb., 196 (2005), 99-134.  doi: 10.1070/SM2005v196n04ABEH000892.  Google Scholar

[30]

P. Sooraksa and G. Chen, Chen system as a controlled weather model –physical principle, engineering design and real applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1830009, 12 pp. doi: 10.1142/S0218127418300094.  Google Scholar

[31] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York, Oxford, 1982.   Google Scholar
[32] A. Tsinober, The Essence of Turbulence as a Physical Phenomenon with Emphasis on Issues of Paradigmatic Nature, Springer Nature Switzerland AG, 2019.  doi: 10.1007/978-3-319-99531-1.  Google Scholar
[33]

V. Yu. Toronov and V. L. Derbov, Boundedness of attractors in the complex Lorenz model, Phys. Rev. E, 55 (1997), 3689-3692.  doi: 10.1103/PhysRevE.55.3689.  Google Scholar

[34]

F. ZhangX. LiaoC. MuG. Zhang and Y.-A. Chen, On global boundedness of the Chen system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 1673-1681.  doi: 10.3934/dcdsb.2017080.  Google Scholar

[35]

F. ZhangX. LiaoG. ZhangC. MuM. Xiao and P. Zhou, Dynamical behaviors of a generalized Lorenz system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 3707-3720.  doi: 10.3934/dcdsb.2017184.  Google Scholar

[36]

F. Zhang and G. Zhang, Boundedness solutions of the complex Lorenz chaotic system, Appl. Math. Comput., 243 (2014), 12-23.  doi: 10.1016/j.amc.2014.05.102.  Google Scholar

[37]

X. Zhang, Dynamics of a class of nonautonomous Lorenz-type systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650208, 12 pp. doi: 10.1142/S0218127416502084.  Google Scholar

[38]

X. Zhang, Dynamics of a class of fractional-order nonautonomous Lorenz-type systems, Chaos, 27 (2017), 041104, 7 pp. doi: 10.1063/1.4981909.  Google Scholar

[39]

X. Zhang, Boundedness of a class of complex Lorenz systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150101, 22 pp. doi: 10.1142/S0218127421501017.  Google Scholar

[40]

Q. ZhaoS. Zhou and X. Li, Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 928-938.  doi: 10.1016/j.cnsns.2006.09.001.  Google Scholar

show all references

References:
[1] C. Bailly and G. Comte-Bellot, Turbulence, Springer International Publishing Switzerland, 2015.  doi: 10.1007/978-3-319-16160-0.  Google Scholar
[2]

R. Barboza, On Lorenz and Chen systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850018, 8 pp. doi: 10.1142/S0218127418500189.  Google Scholar

[3]

R. Barboza and G. Chen, On the global boundedness of the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.  Google Scholar

[4]

S. Celikovsky and G. Chen, Generalized Lorenz systems family revisited, Int. J. Bifurcation Chaos, 31 (2021), 2150079, 15 pp. Google Scholar

[5]

D. Cheban and J. Duan, Recurrent motions and global attractors of nonautonomous Lorenz systems, Dyn. Syst., 19 (2004), 41-59.  doi: 10.1080/14689360310001624132.  Google Scholar

[6]

G. Chen, Generalized Lorenz systems family, preprint, arXiv: 2006.04066, 2020. Google Scholar

[7]

G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.  Google Scholar

[8] P. A. DavidsonY. Kaneda and K. R. Sreenivasan, Ten Chapters in Turbulence, Cambridge Univ. Press, 2013.   Google Scholar
[9]

A. C. FowlerJ. D. Gibbon and M. J. McGuinness, The real and complex Lorenz equations and their relevance to physical systems, Phys. D, 7 (1983), 126-134.  doi: 10.1016/0167-2789(83)90123-9.  Google Scholar

[10]

A. C. FowlerM. J. McGuinness and J. D. Gibbon, The complex Lorenz equations, Phys. D, 4 (1981/82), 139-163.  doi: 10.1016/0167-2789(82)90057-4.  Google Scholar

[11]

M. Franz and M. Zhang, Suppression and creation of chaos in a periodically forced Lorenz system, Phys. Rev. E, 52 (1995), 3558-3565.  doi: 10.1103/PhysRevE.52.3558.  Google Scholar

[12]

H. Haken, Analogy between higher instabilities in fluids and lasers, Phys. Lett. A, 53 (1975), 77-78.  doi: 10.1016/0375-9601(75)90353-9.  Google Scholar

[13]

C. Lainscsek, A class of Lorenz-like systems, Chaos, 22 (2012), 013126, 5 pp. doi: 10.1063/1.3689438.  Google Scholar

[14] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959.   Google Scholar
[15]

G. A. LeonovA. I. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656.  doi: 10.1002/zamm.19870671215.  Google Scholar

[16]

G. A. Leonov and N. V. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen, and Lu systems, Appl. Math. Comput., 256 (2015), 334-343.  doi: 10.1016/j.amc.2014.12.132.  Google Scholar

[17]

C. Letellier, G. F. V. Amaral and L. A. Aguirre, Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models, Chaos, 17 (2007), 023104, 11 pp. doi: 10.1063/1.2645725.  Google Scholar

[18]

D. LiJ. LuX. Wu and G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J. Math. Anal. Appl., 323 (2006), 844-853.  doi: 10.1016/j.jmaa.2005.11.008.  Google Scholar

[19]

X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization (in Chinese), Sci. China Ser. E, 34 (2004), 1404-1419.   Google Scholar

[20]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[21]

J. LüG. ChenD. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X.  Google Scholar

[22]

G. M. MahmoudT. Bountis and E. E. Mahmoud, Active control and global synchronization of the complex Chen and Lü systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4295-4308.  doi: 10.1142/S0218127407019962.  Google Scholar

[23]

J. Pedlosky, Finite-amplitude baroclinic waves with small dissipation, J. Atmos. Sci., 28 (1971), 587-597.  doi: 10.1175/1520-0469(1971)028<0587:FABWWS>2.0.CO;2.  Google Scholar

[24]

J. Pedlosky, The effect of $\beta$ on the chaotic behavior of unstable baroclinic wave, J. Atmos. Sci., 38 (1981), 717-731.   Google Scholar

[25]

W.-X. Qin and G. Chen, On the boundedness of solutions of the Chen system, J. Math. Anal. Appl., 329 (2007), 445-451.  doi: 10.1016/j.jmaa.2006.06.091.  Google Scholar

[26]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

[27]

H. Saberi NikS. Effati and J. Saberi-Nadjafi, New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system, J. Complexity, 31 (2015), 715-730.  doi: 10.1016/j.jco.2015.03.001.  Google Scholar

[28]

R. SaravananO. NarayanK. Banerjee and J. K. Bhattacharjee, Chaos in a periodically forced Lorenz system, Phys. Rev. A, 31 (1985), 520-522.  doi: 10.1103/PhysRevA.31.520.  Google Scholar

[29]

E. A. Sataev, Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Mat. Sb., 196 (2005), 99-134.  doi: 10.1070/SM2005v196n04ABEH000892.  Google Scholar

[30]

P. Sooraksa and G. Chen, Chen system as a controlled weather model –physical principle, engineering design and real applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1830009, 12 pp. doi: 10.1142/S0218127418300094.  Google Scholar

[31] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York, Oxford, 1982.   Google Scholar
[32] A. Tsinober, The Essence of Turbulence as a Physical Phenomenon with Emphasis on Issues of Paradigmatic Nature, Springer Nature Switzerland AG, 2019.  doi: 10.1007/978-3-319-99531-1.  Google Scholar
[33]

V. Yu. Toronov and V. L. Derbov, Boundedness of attractors in the complex Lorenz model, Phys. Rev. E, 55 (1997), 3689-3692.  doi: 10.1103/PhysRevE.55.3689.  Google Scholar

[34]

F. ZhangX. LiaoC. MuG. Zhang and Y.-A. Chen, On global boundedness of the Chen system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 1673-1681.  doi: 10.3934/dcdsb.2017080.  Google Scholar

[35]

F. ZhangX. LiaoG. ZhangC. MuM. Xiao and P. Zhou, Dynamical behaviors of a generalized Lorenz system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 3707-3720.  doi: 10.3934/dcdsb.2017184.  Google Scholar

[36]

F. Zhang and G. Zhang, Boundedness solutions of the complex Lorenz chaotic system, Appl. Math. Comput., 243 (2014), 12-23.  doi: 10.1016/j.amc.2014.05.102.  Google Scholar

[37]

X. Zhang, Dynamics of a class of nonautonomous Lorenz-type systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650208, 12 pp. doi: 10.1142/S0218127416502084.  Google Scholar

[38]

X. Zhang, Dynamics of a class of fractional-order nonautonomous Lorenz-type systems, Chaos, 27 (2017), 041104, 7 pp. doi: 10.1063/1.4981909.  Google Scholar

[39]

X. Zhang, Boundedness of a class of complex Lorenz systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150101, 22 pp. doi: 10.1142/S0218127421501017.  Google Scholar

[40]

Q. ZhaoS. Zhou and X. Li, Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 928-938.  doi: 10.1016/j.cnsns.2006.09.001.  Google Scholar

Table 1.   
$ \widetilde{Z_i} $ $ U_i $ $ U_i-\tfrac{1}{2}\big(\tfrac{\Re\,B-1}{C}- \widetilde{Z_i}\tfrac{\Re\,B}{AC}\big)^2 $
$ i=0 $ $ -\tfrac{37}{12} $ $ \tfrac{5013121}{352800}\approx14.2095 $
$ i=1 $ $ -3.85753 $ $ 14.6399 $ $ 14.2579 $
$ i=2 $ $ -4.73162 $ $ 15.0607 $ $ 14.5737 $
$ i=3 $ $ -5.71849 $ $ 15.4508 $ $ 14.8301 $
$ i=4 $ $ -6.83271 $ $ 15.7805 $ $ 14.9893 $
$ i=5 $ $ -8.09069 $ $ 16.0086 $ $ 14.9999 $
$ i=6 $ $ -9.51099 $ $ 16.0782 $ $ 14.7925 $
$ i=7 $ $ -11.1146 $ $ 15.9126 $ $ 14.2737 $
$ i=8 $ $ -12.925 $ $ 15.4081 $ $ 13.3189 $
$ i=9 $ $ -14.9691 $ $ 14.426 $ $ 11.7629 $
$ i=10 $ $ -17.277 $ $ 12.7821 $ $ 9.38745 $
$ i=11 $ $ -19.8826 $ $ 10.2326 $ $ 5.90533 $
$ i=12 $ $ -22.8245 $ $ 6.45598 $ $ 0.940016 $
$ i=13 $ $ -26.1459 $ $ 1.03054 $ $ -6.00074 $
$ \widetilde{Z_i} $ $ U_i $ $ U_i-\tfrac{1}{2}\big(\tfrac{\Re\,B-1}{C}- \widetilde{Z_i}\tfrac{\Re\,B}{AC}\big)^2 $
$ i=0 $ $ -\tfrac{37}{12} $ $ \tfrac{5013121}{352800}\approx14.2095 $
$ i=1 $ $ -3.85753 $ $ 14.6399 $ $ 14.2579 $
$ i=2 $ $ -4.73162 $ $ 15.0607 $ $ 14.5737 $
$ i=3 $ $ -5.71849 $ $ 15.4508 $ $ 14.8301 $
$ i=4 $ $ -6.83271 $ $ 15.7805 $ $ 14.9893 $
$ i=5 $ $ -8.09069 $ $ 16.0086 $ $ 14.9999 $
$ i=6 $ $ -9.51099 $ $ 16.0782 $ $ 14.7925 $
$ i=7 $ $ -11.1146 $ $ 15.9126 $ $ 14.2737 $
$ i=8 $ $ -12.925 $ $ 15.4081 $ $ 13.3189 $
$ i=9 $ $ -14.9691 $ $ 14.426 $ $ 11.7629 $
$ i=10 $ $ -17.277 $ $ 12.7821 $ $ 9.38745 $
$ i=11 $ $ -19.8826 $ $ 10.2326 $ $ 5.90533 $
$ i=12 $ $ -22.8245 $ $ 6.45598 $ $ 0.940016 $
$ i=13 $ $ -26.1459 $ $ 1.03054 $ $ -6.00074 $
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